D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras

0.46 MB·English

by  E. Frenkel

D–MODULES ON THE AFFINE GRASSMANNIAN AND
REPRESENTATIONS OF AFFINE KAC-MOODY ALGEBRAS
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2 EDWARDFRENKELANDDENNISGAITSGORY

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Dedicated toVictorKac on his 60th birthday
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1. Introduction

] 1.1. Let g be a simple Lie algebraoverC, and G the correspondingalgebraicgroupof adjoint
G
type. Givenaninvariantinnerproductκong,letˆg denotethecorrespondingcentralextension
κ
A
of the formal loop algebra g⊗C((t)), called the affine Kac-Moody algebra ˆg ,
κ
.
h 0→C1→ˆg →g⊗C((t))→0,
κ
t
a
with the two-cocycle defined by the formula
m
x⊗f(t),y⊗g(t)7→−κ(x,y)·Res fdg.
[ t=0

Denotebyˆg –modthecategoryofˆg -moduleswhicharediscrete,i.e.,anyvectorisannihilated
3 κ κ
v by the Lie subalgebra g⊗tNC[[t]] for sufficiently large N ≥ 0, and on which 1 ∈ C ⊂ gˆκ acts
3 as the identity. We will refer to objects of these category as modules at level κ.
7 Let Gr = G((t))/G[[t]] be the affine Grassmannian of G. For each κ there is a category
G
1
D (Gr )–mod of κ-twistedright D-modules on Gr (see [BD]). We have the functor of global
3 κ G G
sections
0
3 Γ:Dκ(GrG)–mod→ˆgκ–mod, F 7→Γ(GrG,F).
0 Let κ be the Killing form, κ (x,y) = Tr(ad (x)◦ad (y)). The level κ = −1κ
/ Kil Kil g g crit 2 Kil
h is called critical. A level κ is called positive (resp., negative, irrational) if κ = c·κKil and
t c+ 1 ∈Q>0 (resp., c+ 1 ∈Q<0, c∈/ Q).
a 2 2
m Itisknownthatthefunctorofglobalsectionscannotbeexactwhenκispositive. Incontrast,
when κ is negative or irrational, the functor Γ is exact and faithful, as shown by A. Beilinson
:
v and V. Drinfeld in [BD], Theorem 7.15.8. This statement is a generalizationfor affine algebras
i
X of the famous theorem of A. Beilinson and J. Bernstein, see [BB], that the functor of global
sections fromthe categoryof λ-twistedD-modules onthe flag variety G/B is exactwhen λ−ρ
r
a is anti-dominant and it is faithful if λ−ρ is, moreover,regular.
Thepurposeofthispaperistoconsiderthefunctorofglobalsectionsinthecaseofthecritical
level κ . (In what follows we will slightly abuse the notation and replace the subscript
crit κcrit
simply by .) Unfortunately, it appears that the approach of [BD] does not extend to the
crit
critical level case, so we have to use other methods to analyze it. Our main result is that the
functor of global sections remains exact at the critical level:
Theorem 1.2. The functor Γ:D (Gr )–mod→ˆg –mod is exact.
crit G crit
Inotherwords,weobtainthatfor anyobjectF ofD (Gr )–modwehaveHi(Gr ,F)=0
crit G G
for i > 0. Moreover, we will show that if F 6= 0, then H0(Gr ,F) = Γ(Gr ,F) 6= 0. This
G G
property is sometimes referred to as “D-affineness” of Gr .
G
Date:November2003.
1
2 EDWARDFRENKELANDDENNISGAITSGORY
In fact, we will prove a stronger result. Namely, we note after [BD], that for a critically
twisted D-module F on Gr , the action of ˆg on Γ(Gr ,F) extends to an action of the
G crit G
renormalized enveloping algebra Uren(ˆg ) of Sect. 5.6 of loc.cit. Following a conjecture
crit
and suggestion of Beilinson, we show that the resulting functor from D (Gr )–mod to the
crit G
category of Uren(ˆg )-modules is fully-faithful.
crit
1.3. OurmethodofproofofTheorem1.2usesthechiralalgebraofdifferentialoperatorsD
G,κ
introduced in [AG]. Modules over D should be viewed as (twisted) D-modules on the loop
G,κ
group G((t)). In particular, the category of κ-twisted D-modules on Gr is equivalent to the
G
subcategoryinD –mod,consistingofmodules,whichareintegrablewithrespecttotheright
G,κ
actionofg[[t]](seeTheorem2.5). ThefunctorofglobalsectionsonGr corresponds,underthis
G
equivalence, to the functor of g[[t]]-invariants. Therefore, we need to prove that this functor of
invariants is exact.
This approach may be applied both when the level κ is negative (or irrational) and critical.
In the case of the negative or irrational level the argument is considerably simpler, and so we
obtain a proof of the exactness of Γ, which is different from that of [BD] (see Sect. 2).
The argument that we use for affine Kac-Moody algebras yields also a different proof of the
exactness statement from [BB]. Let us sketch this proof. For a weight λ, let Dλ(G/B)–mod
be the category of left λ-twisted D-modules on G/B (here for an integral λ, by a λ-twisted D-
module on G/B we understand a module over the sheaf of differential operators acting on the
line bundleG× λ). Letπ denotethe naturalprojectionG→G/B,andobservethatthepull-
B
back functor (in the sense of quasicoherent sheaves) lifts to a functor π∗ : Dλ(G/B)–mod →
D(G)–mod. Furthermore, for a D-module F′ on G, the space of its global sections Γ(G,F′) is
naturally a bimodule over g due to the action of G on itself by left and the right translations.
For F ∈Dλ(G/B)–mod we have
Γ(G/B,F)≃Hom C−λ,Γ(G,π∗(F)) ,
b
(cid:0) (cid:1)
where b is the Borel subalgebra of g, C−λ its one-dimensional representation corresponding
to weight −λ, and Γ(G,π∗(F)) is a b-module via b ֒→ g and the right action of g. But the
g-module Γ(G,F′), where F′ =π∗(F) (with respectthe rightg-action),belongsto the category
O. Thus, we obtain a functor
Γ′ :Dλ(G/B)−mod→O, F 7→Γ(G,π∗(F)),
and
Γ(G/B,F)≃HomO(M(−λ),Γ′(F)),
where M(−λ) is the Verma module with highest weight −λ.
The functor Γ′ is exact because G is affine, and it is well-known that M(µ) is a projective
objectofOpreciselywhenµ+ρis dominant. Hence Γ isthe compositionoftwo exactfunctors
and, therefore, is itself exact.
ThisreprovestheBeilinson-Bernsteinexactnessstatement. Note,however,thatthemethods
described above do not give the non-vanishing assertion of [BB].
1.4. The proof of the exactness result in the negative (or irrational) level case is essentially a
wordforwordrepetitionoftheaboveargument,onceweareabletomakesenseofthecategory
of D-modules on G((t)) as the category of D -modules. The key fact that we will use will be
G,κ
the same: that the corresponding vacuum Weyl module Vg,κ′ is projective in the appropriate
category O if κ′ is positive or irrational.
This argument does not work at the critical level, because in this case the corresponding
Weyl module V is far from being projective in the category ˆg −mod. Roughly, the
g,crit crit
D-MODULES ON THE AFFINE GRASSMANNIAN 3
picture is as follows. Modules over gˆ give rise to quasicoherent sheaves over the ind-scheme
crit
Spec(Z ), where Z is the center of the completed universalenveloping algebraof ˆg (this
g,x g,x crit
istheind-schemeofLg-opersonthepunctureddisc,whereLgistheLanglandsdualLiealgebra
to g). The ind-scheme Spec(Z ) contains a closedsubscheme Spec(z ) (this is the scheme of
g,x g,x
Lg-opers on the disc). The module V is supported on Spec(z ) and is projective in the
g,crit g,x
category of gˆ -modules, which are supported on Spec(z ) and are G(Oˆ )-integrable.
crit g,x x
The problem is, however, that the ˆg -modules of the form Γ(G((t)),π∗(F)), where π
crit
is the projection G((t)) → G((t))/G[[t]] ≃ Gr , are never supported on Spec(z ). There-
G g,x
fore we need to show that the functor of taking the maximal submodule of Γ(G((t)),π∗(F)),
which is supported on Spec(z ), is exact. We do that by showing that the action of ˆg on
g,x crit
Γ(G((t)),π∗(F)) automaticallyextends to the actionof the renormalizedchiralalgebraAren,τ,
g
which is closely related to the renormalized enveloping algebra Uren(ˆg ), mentioned above.
crit
Consider the following analogy. Let X be a smooth variety and Y its smooth closed sub-
variety. Then we have a natural functor, denoted i!, from the category of O -modules, set-
X
theoreticallysupportedonY,tothe categoryofO -modules: thisfunctortakesanO -module
Y X
F toitsmaximalsubmodulesupportedscheme-theoreticallyonY. Thisisnotanexactfunctor.
ButthecorrespondingfunctorfromthecategoryofrightD-modulesonX,alsoset-theoretically
supportedonY,tothecategoryofrightD-modulesonY isexact,accordingtoabasictheorem
due to Kashiwara.
In our situation the role of the category of O -modules is played by the category ˆg −
X crit
mod, and the role of the category of D-modules is played by the category of modules over the
chiral algebra Aren,τ. We show that the above functor of taking the maximal submodule of
g
Γ(G((t)),π∗(F)), which is supported on Spec(z ), factors through the latter category, and
g,x
this allows us to prove the required exactness.
1.5. Contents. Let us briefly describe how this paper is organized. In Sect. 2 we treat the
negative level case. In Sect. 3 we recall some facts about commutative D-algebras and the
description of the center of the Kac-Moody chiral algebra at the critical level. In Sect. 4 we
discuss several versions of the renormalized universal enveloping algebra at the critical level in
the setting of chiral algebras. In Sect. 5 we study the chiral algebra of differential operators
D when κ = κ . In Sect. 6 we derive our main Theorem 1.2 from two other statements,
G,κ crit
Theorems 6.11 and 6.15. In Sect. 7 we prove Theorem 6.15, generalizing Kashiwara’s theorem
aboutD-modulessupportedonasubvariety. InSect.8weproveTheorem6.11anddescribethe
categoryofgˆ -modules,whicharesupportedonSpec(z )andareG(Oˆ )-integrable. Finally,
crit g,x x
in Sect. 9 we prove that the functor Γ is faithful.
1.6. Conventions. Our basic tool in this paper is the theory of chiral algebras. The founda-
tional work [CHA] on this subject will soon be published (in our references we use the most
recent version; a previous one is currently available on the Web). In addition, an abridged
summaryofthe resultsof[CHA]thatareusedinthis papermaybe found in[AG]. We wishto
remarkthatallchiralalgebrasconsideredinthispaperareuniversalinthesensethattheycome
from quasi-conformal vertex algebras by a construction explained in [FB], Ch. 18. Therefore
all results of this paper may be easily rephrased in the language of vertex algebras. We have
chosenthe languageofchiralalgebrasinordertobe consistentwiththe languageusedin[AG].
Wealsousesometheresultsfrom[BD],whichisstillunpublished,butavailableontheWeb.
The notation in this paper mainly follows that of [AG]. Throughout the paper, X will be a
fixed smooth curve; we will denote by O (resp., ω , T and D ) its structure sheaf (resp.,
X X X X
the sheaf of differentials, the tangent sheaf and the sheaf of differential operators).
4 EDWARDFRENKELANDDENNISGAITSGORY
We will work with D-modules on X, and in our notation we will not distinguish between
left and right D-modules, i.e., we will denote by the same symbol a left D-module M and the
corresponding right D-module M⊗ω . The operations of tensor product, taking symmetric
X
algebra, and restriction to a subvariety must be understood accordingly.
We will denote by ∆ the diagonal embedding X → X ×X, and by j the embedding of its
complement X ×X −∆(X)→X ×X. If x ∈X is a point, we will often consider D-modules
supported at x. In this case, our notation will not distinguish between such a D-module and
the underlying vector space.
We will use the notation A×C for a fiber product of A and C over B, and the notation
B
P× V for the twist of a G-module V by a G-torsor P.
G
Finally, if C is a category and C is an object of C, we will often write C ∈C.
1.7. Acknowledgments. D.G. would like to express his deep gratitude to A. Beilinson for
explainingtohimthe theoryofchiralalgebras,aswellasfornumerousconversationsrelatedto
this paper. He would also like to thank S. Arkhipov, J. Bernstein for stimulating and helpful
discussions.
Inaddition,bothauthorswouldliketothankA.Beilinsonforhelpfulremarksandsuggestions
and B. Feigin for valuable discussions.
The research of E.F. was supported by grants from the Packard foundation and the NSF.
D.G. is a long-term prize fellow of the Clay Mathematics Institute.
2. The case of affine algebras at the negative and irrational levels
2.1. In this section we will show that the functor of global sections
Γ:D (Gr )−mod→ˆg −mod
κ G κ
is exact when κ is negative or irrational. A similar result has been proved by Beilinson and
Drinfeld in [BD], Theorem 7.15.8, by other methods. The setting of [BD] is slightly different:
they consider twisted D-modules on the affine flag variety Fl = G((t))/I instead of Gr =
G G
G((t))/G[[t]], where I ⊂ G[[t]] is the Iwahori subgroup, i.e., the preimage of a fixed Borel
subgroup B ⊂ G under the projection G[[t]] → G. Here is the precise statement of their
theorem:
Recall that for any affine weight λˆ = (λ,2hˇ ·c) (where λ is a weight of g, c ∈ C and hˇ is
the dual Coxeter number), we can consider the correspondingcategoryD (Fl )–mod, of right
λˆ G
λˆ-twisted D-modules on Fl . A weight λˆ is called anti-dominant if the corresponding Verma
G
module M(λˆ) over ˆg (where κ = c·κ ) is irreducible. According to a theorem of Kac and
κ Kil
Kazhdan(see[KK]),thisconditioncanbeexpressedcombinatoriallyashλˆ+ρ ,αˇ i∈/ Z>0,
aff aff
where α runs over the set of all positive affine coroots. We have:
aff
Theorem2.2. Ifλˆ isanti-dominant,thenthefunctorofsectionsΓ:D (Fl )–mod→ˆg –mod
λˆ G κ
is exact.
Theorem2.2formallyimpliestheexactnessstatementonGr (i.e.,Theorem2.4below)only
G
for κ = c·κ with c either irrational, or c+ 1 < −1+ 1 ; so our exactness result is slightly
Kil 2 2hˇ
sharper than that of [BD]. The proof of Theorem 2.4 given below can be extended in a rather
straightforward way to reprove Theorem 2.2. In contrast, in the case of the critical level, it
is essential that we consider D-modules on Gr and not on Fl ; in the latter case the naive
G G
analogue of the exactness statement is not true.
Finally, note that Theorem 7.15.8 of [BD] contains also the assertion that for 0 6= F ∈
D (Fl )–mod, then the space of sections Γ(Gr ,F) is non-zero, implying a similar statement
λ G G
e
D-MODULES ON THE AFFINE GRASSMANNIAN 5
forF ∈D (Gr ). InSect.9,wewillreprovethisfactaswell,byadifferentmethod. Thisproof
κ G
is the same in the negative and the critical level cases.
2.3. Thus, our goal in this section is to prove the following theorem:
Theorem 2.4. The functor Γ : D (Gr )–mod → ˆg –mod is exact when κ is negative or
κ G κ
irrational.
The starting point of our proof is the following. Recall the chiral algebra D (on our
G,κ
curve X), introduced in [AG]. Let D –mod denote the category of chiral D -modules
G,κ G,κ
concentrated at a point x ∈ X. In [AG] it was shown that D –mod is a substitute for the
G,κ
category of twisted D-modules on the loop group G((t)), where t is a formal coordinate on X
near x.
In particular, we have the forgetful functor
D –mod→(ˆg ×gˆ )–mod,
G,κ κ 2κcrit−κ
whereˆg –mod(resp.,ˆg –mod)isthe categoryofrepresentationsofthe affinealgebraat
κ 2κcrit−κ
the level κ (resp., 2κ −κ). This functor corresponds to the action of the Lie algebra g((t))
crit
onG((t)) by left andrighttranslations. In what follows,for a module M∈D –mod, we will
G,κ
refer to the corresponding actions of ˆg and ˆg on it as “left” and “right”, respectively.
κ 2κcrit−κ
LetO ≃C[[t]]be the completedlocalringatx. Considerthe subalgebrag(O )⊂ˆg .
x x 2κcrit−κ
Let ˆg –modG(Ox) be the subcategory of gˆ –mod whose objects are the ˆg -
2κbcrit−κ 2κcrit−κ b 2κcrit−κ
modules, onwhichthbe actionofg(O ) maybe exponentiatedto anactionofthe corresponding
x
group G(O ). Let D –modG(Ox) denote the full subcategory of D –mod whose objects
x G,κ b G,κ
belong tobˆg2κcrit−κ–modG(Ox) unbder the right action of gˆ2κcrit−κ–mod.
The following result hasbbeen established in [AG]:
Theorem 2.5. There exists a canonical equivalence of categories
D (Gr )–mod≃D –modG(Ox).
κ G G,κ
b
If F is an object of Dκ(GrG)–mod, and MF the corresponding object of DG,κ–modG(Ox), then
the ˆgκ-module Γ(GrG,F) identifies with (MF)g(Ox), the space of invariants in MF witbh respect
to the Lie subalgebra g(O )⊂gˆ under thbe right action.
x 2κcrit−κ
2.6. To prove the exacbtness of the functor Γ : D (Gr )–mod → ˆg –mod, for negative or
κ G κ
irrational κ, we compose it with the tautological forgetful functor ˆg –mod →Vect. By Theo-
κ
rem 2.5, this composition can be rewritten as
D –modG(Ox) →ˆg –modG(Ox) →Vect,
G,κ 2κcrit−κ
b b
where the first arrow is the forgetful functor, and the second arrow is M7→Mg(Ox).
For an arbitrary level κ′, let Vg,κ′ be the vacuum Weyl module, i.e., b
Vg,κ′ ≃Indggˆκ(O′x)⊕C1(C),
whereg(Ox)actsonCbyzeroand1actsastheibdentity. Tautologically,foranyM∈ˆgκ′–mod,
we have:
b
(2.1) Homgˆκ′(Vg,κ′,M)≃Mg(Ox).
b
Moreover,Vg,κ′ is G(Ox)-integrable, i.e., belongs to ˆgκ′–modG(Ox).
b
b
6 EDWARDFRENKELANDDENNISGAITSGORY
Observe that the condition that κ is negative or irrational is equivalent to κ′ := 2κ −κ
crit
being positiveorirrational. Therefore,toprovethe exactnessofΓ,itisenoughtoestablishthe
following:
Proposition2.7. Ifκ′ ispositiveorirrational, themoduleVg,κ′ isprojectiveinˆgκ′–modG(Ox).
b
This proposition is well-known, and the proof is based on considering eigenvalues of the
Segal-Sugawaraoperator L . We include the proof for completeness.
0
Proof. Recall that for every non-critical value of κ′, the vector space underlying every object
M∈ˆgκ′–modcarriesacanonicalendomorphismL0 obtainedviathe Segal-Sugawaraconstruc-
tion, such that the action of ˆgκ′ commutes with L0 in the following way:
(2.2) [L ,x⊗tn]=−n·x⊗tn, x∈g,n∈Z.
0
Explicitly, let {xa,x } be bases in g, dual with respect to κ . The operator
a Kil
(2.3) S = xa·x +2 xa⊗t−n·x ⊗tn
0 a a
Xa Xa nX>0
is well-defined onevery object of ˆgκ′–mod, and it has the following commutationrelationwith
elements of ˆgκ′:
(2.4) [S ,x⊗tn]=−(2c′+1)·n·x⊗tn, x∈g,n∈Z,
0
where c′ is such that κ′ = c′·κ . Therefore, for c′ 6= −1, the operator L := 1 ·S has
Kil 2 0 2c′+1 0
the required properties.
Foranintegraldominantweightλofg,letVλ bethefinite-dimensionalirreducibleg-module
with highest weight λ and Vλg,κ′ the corresponding Weyl module over ˆgκ′,
Vλg,κ′ =Indggˆκ(O′x)⊕C1(Vλ),
b
whereg(O )actsonVλ throughthehomomorphismg(O )→gand1actsastheidentity. Then
x x
we find from formula (2.3) that L acts on the subspace Vλ ⊂Vλ by the scalar Cg(λ), where
b 0 b g,κ′ 2c′+1
C (λ) is the scalar by which the Casimir element xa ·x of U(g) acts on Vλ. Note that
g a a
Cg(λ) is a non-negative rational number for any doPminant integral weight λ, and Cg(λ)6=0 if
λ6=0.
Since Vλg,κ′ is generated from Vλ by the elements x⊗tn ∈ ˆgκ′, n < 0, we obtain that the
action of L0 on Vλg,κ′ is semi-simple. Moreover, since every object M ∈ ˆgκ′–modG(Ox) has a
filtration whose subquotients are quotients of the Vλ ’s, the action of L on any sbuch M is
g,κ′ 0
locally-finite.
Suppose now that we have an extension
(2.5) 0→M→M→Vg,κ′ →0
in gˆκ′–modG(Ox). Let v0 ∈M be a lift to M eof the generating vector v0 ∈Vg,κ′. Without loss
of generality wbe may assume that v0 has the same generalized eigenvalue as v0, i.e., 0, with
respectto the actionofeL . Iet is sufficientteo show thatthen v0 belongsto (M)g⊗tC[[t]]. Indeed,
0
e
if this is so, then v0 is annihilated by the entire Lie subalgebra g(O ), due to the eigenvalue
e x e
condition, which would mean that the extension (2.5) splits.
e b
D-MODULES ON THE AFFINE GRASSMANNIAN 7
Suppose that this is not the case, i.e., that v0 is not annihilated by g⊗tC[[t]]. Then we can
find a sequence of elements xi ⊗tni ∈ g⊗tC[[t]], which we can assume to be homogeneous,
automatically of negative degrees with respecteto L , such that the vector
0
w =x1⊗tn1 ·...·xk⊗tnk ·v0 ∈M
is non-zero and is annihilated by g⊗tC[[t]]. But then, on the one hand, the eigenvalue of L
e 0
on w is deg(x1⊗tn1)+...+deg(xk⊗tnk)=−(n1+...+nk)∈Z<0, but on the other hand, it
must be of the form Cg(λ), which is not in Q<0, by our assumption.
c′+1
2 (cid:3)
3. Center of the Kac-Moody chiral algebra at the critical level
3.1. Let A be a unital chiral algebra on X. In what follows we will work with a fixed point
x∈X and denote by A–mod the category of chiral A-modules, supported at x.
Recall that the center of A, denoted by z(A), is by definition the maximal D-submodule of
A for, which the Lie-* bracket z(A)⊠A → ∆(A) vanishes. It is easy to see that z(A) is a
!
commutative chiral subalgebra of A. For example, the unit ω ֒→ A is always contained in
X
z(A).
LetA bethe chiraluniversalenvelopingalgebraoftheLie-*algebraL :=g⊗D ⊕ω
g,κ g,κ X X
at the level κ (modulo the relation equating the two embeddings of ω ). We have the basic
X
equivalence of categories:
A –mod≃ˆg –mod.
g,κ κ
It is well-known that when κ 6= κ , the inclusion ω → z(A ) is an isomorphism. Let
crit X g,κ
us denote by z the commutative chiral algebra z(A ). In Theorem 3.4 below we will recall
g g,crit
the description of z obtained in [FF, F].
g
Let z be the fiber of z at x; this is a commutative algebra. We have the natural maps
g,x g
zg,x −→(Vg,crit)g(Ox) ←∼−Endˆgcrit(Vg,crit),
b
where the left arrow is obtained from the definition of the center of a chiral algebra, and the
rightarrowassignsto e∈End (V )the vectore·v0, wherev0 is the canonicalgenerator
gˆcrit g,crit
of V .
g,crit
The resulting homomorphism of algebras z → End (V ) is an isomorphism. In
g,x gˆcrit g,crit
fact, for any chiral algebra A, its center z(A) identifies with the D-module of endomorphisms
of A regarded as a chiral A-module. At the level of fibers, we have a map in one direction
z(A)x →EndA–mod(Ax). Thismapisanisomorphismifacertainflatnessconditionissatisfied.
This condition is always satisfied if A is “universal”, i.e., comes from a quasi-conformalvertex
algebra, which is the case of A .
g,crit
3.2. For a chiral algebra A, let Aˆ be the canonical topological associative algebra attached
x
to the point x, see [CHA], Sect. 3.6.2. By definition, the category A–mod endowed with the
tautological forgetful functor to the category of vector spaces, is equivalent to the category of
discrete continuous Aˆ -modules, denoted Aˆ –mod.
x x
Forexample,whenA=A ,thecorrespondingalgebraAˆ identifieswiththecompleted
g,κ g,κ,x
universalenvelopingalgebraofˆg modulotherelation1=1. WedenotethisalgebrabyU′(ˆg ).
κ κ
WhenA=Biscommutative,thealgebraBˆ iscommutativeaswell,see[CHA],Sects. 3.6.6
x
and 2.4.8. In fact, Bˆ can be naturally represented as limBi, where Bi are subalgebras of
x x
←−
B, such that Bi| ≃ B| . In particular, we have a surjective homomorphism Bˆ → B ;
X−x X−x x x
8 EDWARDFRENKELANDDENNISGAITSGORY
the subcategory B –mod ⊂ Bˆ –mod is the full subcategory of B–mod, whose objects are
x x
central B-modules, supported at x ∈ X. (Recall that a B-module M is called central if the
action map j j∗(B ⊠ M) → ∆(B) comes from a map B ⊗ M → M, i.e., factors through
∗ !
j j∗(B⊠M)։∆(B⊗M).)
∗ !
We will view Spec(Bˆ ) as an ind-scheme lim Spec(Bi); we have a closed embedding
x x
−→
Spec(B )֒→Spec(Bˆ ).
x x
By taking B = z , we obtain a topological commutative algebra ˆz , which we will also
g g,x
denote by Z . The corresponding map Spec(z )֒→Spec(Z ) will be denoted by ı.
g,x g,x g,x
For any chiral algebra A we have a homomorphism
z(A) →Z(Aˆ ),
x x
where Z(Aˆ ) is the centerofAˆ . We dodnot knowwhether this mapis alwaysanisomorphism,
x x
but can show that it is an isomorphism for A = A , using the description of z , given by
g,crit g
Theorem 3.4(1) below (see [BD], Theorem 3.7.7). In other words, Z maps isomorphically to
g,x
the center of U′(ˆg ).
crit
3.3. Letusrecalltheexplicitdescriptionofz andZ dueto[FF,F]. LetLGbethealgebraic
g g,x
group of adjoint type whose Lie algebra is the Langlands dual to g. Denote by Op (D ) the
LG x
affine scheme of LG-opers on the disc D = Spec(O ). These are triples (F,F ,∇), where F
x x B
is a LG–torsor over D , F is its reduction to a fixed Borel subgroup LB ⊂ LG and ∇ is a
x B b
connectiononF (automatically flat)suchthat F and∇ arein aspecialrelativeposition (see,
B
e.g., [F] for details).
There exists an affine D -scheme J(Op (X)) of jets of opers on X, whose fiber at x∈X
X LG
is Op (D ) (see [BD], Sect. 3.3.3), and so the corresponding sheaf of algebras of functions
LG x
Fun(J(Op (X))) on X is a commutative chiral algebra. (In what follows, Fun(Y) stands for
LG
the ring of regular functions on a scheme Y.)
The canonicaltopologicalalgebraassociatedto Fun(J(Op (X)))atthe pointx is nothing
LG
but the topological algebra of functions on the ind-affine space Op (D×) of LG-opers on the
LG x
punctured disc D× = Spec(K ), where K is the field of fractions of O . The following was
x x x x
established in [FF, F]:
b b b
Theorem 3.4.
(1) There exists a canonical isomorphism of D -algebras
X
z ≃Fun(J(Op (X))).
g LG
In particular, we have an isomorphism of commutative algebras z ≃Fun(Op (D )) and of
g,x LG x
commutative topological algebras Z ≃Fun(Op (D×)).
g,x LG x
(2) On the associated graded level, we have a commutative diagram of isomorphisms:
gr(z ) ←−−−− gr Fun(Op (D ))
g,x LG x
(cid:0) (cid:1)
 
Fun (g∗×Gm Γy(Dx,ΩX))G(Ox) ←−−−− Fun (Lg/LG)×yGm Γ(Dx,ΩX) ,
(cid:16) b (cid:17) (cid:0) (cid:1)
where Lg/LG=Spec(Fun(Lg)LG).
D-MODULES ON THE AFFINE GRASSMANNIAN 9
Note that in the lower left corner of the above commutative diagram we have used the
identification gr(Vg,crit)≃Sym g⊗(Kx/Ox) ≃Fun(g∗×Gm Γ(Dx,ΩX)), and
(cid:16) (cid:17)
g∗/G≃bh∗/Wb ≃Lh/W ≃Lg/LG.
3.5. To proceed we need to recall some more material from [CHA] about commutative D-
algebras (which, according to our conventions, we do not distinguish them from commutative
chiral algebras).
IfBisacommutativeD -algebra,considerthe B-module Ω1(B)of(relativewithrespectto
X
X) differentials on B, i.e., Ω1(B)≃I /I2, where I is the kernelof the product B ⊗ B→B.
B B B
O
X
From now on we will assume that B is finitely generated as a D -algebra; in this case Ω1(B)
X
is finitely generated as a B⊗D -module.
X
RecallthatgeometricpointsoftheschemeSpec(B )(resp.,oftheind-schemeSpec(Bˆ ))are
x x
thesameashorizontalsectionsofSpec(B)overtheformaldiscD (resp.,theformalpunctured
x
disc D×), see [CHA], Sect. 2.4.9. Let us explain the geometric meaning of Ω1(B) in terms of
x
these identifications.
Letz beapointofSpec(B ),correspondingtoahorizontalsectionφ :O →B . Evidently,
x z x x
we have: φ∗(Ω1(B)) ≃T∗(Spec(B )), where T∗ denotes the cotangent space at z.
z x z x z b
From the definition of Bˆ we obtain a map
x
(3.1) H0 D×,φ∗(Ω1(B)) →T∗(Spec(Bˆ )).
DR x z z x
(Since the D-module φ∗(Ω1(B))(cid:0)on D is finitely(cid:1)generated, its de Rham cohomology over the
z x
formalandformalpunctures disc makes obvioussense.) One canshow that the mapof (3.1) is
actually an isomorphism.
From the short exact sequence
(3.2) 0→H0 (D ,φ∗(Ω1(B)))→H0 (D×,φ∗(Ω1(B)))→φ∗(Ω1(B)) →0,
DR x z DR x z z x
we obtain also an identification
H0 (D ,φ∗(Ω1(B)))≃N∗(B ),
DR x z z x
where N∗(B ) denotes the conormal to Spec(B ) inside Spec(Bˆ ) at the point z.
z x x x
Assume now that B is smooth (see [CHA], Sect. 2.3.15for the definition of smoothness). In
this case Ω1(B) is a finitely generated projective B⊗D -module.
X
Consider the dual of Ω1(B), i.e.,
Θ(B):=HomB⊗DX Ω1(B),B⊗DX .
ThisisacentralB-module,calledthetangentmo(cid:0)duletoB. More(cid:1)over,Θ(B)carriesacanonical
structure of Lie-* algebroid over B (see below). Evidently, Θ(B) is also projective and finitely
generated as a B⊗D -module.
X
By dualizing the members of the short exact sequence (3.2), we obtain the identifications
(cf. [CHA], Sect. 2.5.21):
H0 (D ,φ∗(Θ(B)))≃T (Spec(B )), H0 (D×,φ∗(Θ(B)))≃T (Spec(Bˆ )),
DR x z z x DR x z z x
and φ∗(Θ(B)) ≃N (B ).
z x z x
The next definition will be needed in Sect. 6. Let I denote the kernel Bˆ → B . The
x x
quotient I/I2 is a topological module over B , and the normal bundle, N(B ), to Spec(B )
x x x
inside Spec(Bˆ ) can always be defined as the group ind-scheme Spec(Sym (I/I2)). Let now
x B
x
E⊂N(B ) be a group ind-subscheme, and let E⊥ be its annihilator in I/I2.
x
10 EDWARDFRENKELANDDENNISGAITSGORY
WeintroducethesubcategoryBˆx–modE insidethecategoryBˆx–modofallchiralB-modules
supported at x by imposing the following two conditions:
(1)We requirethata moduleM,viewedasaquasicoherentsheafonSpec(Bˆ ), issupportedon
x
theformalneighborhoodofSpec(B ). Inparticular,Macquiresacanonicalincreasingfiltration
x
M= ∪ M , where M ⊂M is the submodule consisting of sections annihilated by Ii.
i i
i≥1
(2) We require that the natural map I/I2 ⊗ M /M →M /M vanish on E⊥ ⊂I/I2.
i+1 i i i−1
B
x
Note that the category Bˆx–modE is in general not abelian.
3.6. Let us now recall the notion of Lie-* algebroid over a commutative D algebra B (cf.
X
[CHA], Sect. 2.5).
Let L be a central B-module. A structure of a Lie-* algebroid over B on L is the data of
a Lie-* bracket L⊠L →∆(L) and an action map L⊠B → ∆(B), which satisfy the natural
! !
compatibility conditions given in [CHA], Sect. 1.4.11 and 2.5.16.
If B is smooth, then Θ(B) is well-defined, and it carries a canonical structure of Lie-*
algebroid over B. It is universal in the sense that for any Lie-* algebroid L, its action on B
factors through a canonical map of Lie-* algebroids ̟ :L→Θ(B), called the anchor map.
RecallnowthatastructureonBofchiral-Poisson(or,coisson,inthe terminologyof[CHA])
algebraisaLie-*bracket(calledchiral-Poissonbracket)B⊠B→∆(B), satisfyingthe Leibniz
!
rule with respect to the multiplication on B (cf. [CHA], Sect. 1.4.18 and 2.6.).
If B is a chiral-Poisson algebra, Ω1(B) acquires a unique structure of Lie-* algebroid, such
that the de Rham differential d:B→Ω1(B) is a map of Lie-* algebras,and the composition
B⊠Bd−×→idΩ1(B)⊠B→∆(B)
!
coincides with the chiral-Poissonbracket.
Following [CHA], Sect. 2.6.6, we call a chiral-Poisson structure on B elliptic if (a) B is
smooth, (b) the anchor map ̟ : Ω1(B) → Θ(B) is injective, and (c) coker(̟) is a projective
B-module of finite rank.
3.7. Finally, let us recallthe definition ofthe chiral-Poissonstructure on z . Considerthe flat
g
C[[ℏ]]-family of chiral algebras Ag,ℏ, corresponding to the pairing κℏ =κcrit+ℏ·κ0, where κ0
is an arbitrary fixed non-zero invariant inner product.
Fortwosectionsa,b∈zg,considertwoarbitrarysectionsaℏ,bℏ ∈Ag,ℏ,whosevaluesmodulo
ℏ area andb respectively,and consider[aℏ,bℏ]∈∆!(Ag,ℏ). By assumption, the lastexpression
vanishes modulo ℏ. Therefore the section ℏ1[aℏ,bℏ] ∈ ∆!(Ag,ℏ) is well-defined. Moreover, its
value mod ℏ does not depend on the choice of aℏ and bℏ.
Therefore we obtain a map
1
a,b∈zg 7→ ℏ[aℏ,bℏ] mod ℏ∈∆!(Ag,crit),
and it is easy to see that its image belongs to ∆(z ). Furthermore, it is straightforward to
! g
verifythattheresultingmapz ⊠z →∆(z )satisfiestheaxiomsofthechiral-Poissonbracket,
g g ! g
see [CHA], Sect. 2.7.1.
LetusnowdescribeintermsofTheorem3.4abovetheLie-*algebroidΩ1(z ),resultingfrom
g
the chiral-Poissonstructure on z .
g
First, recall from [CHA], Sect. 2.4.11, that if M is a central module over a commutative
chiral algebra B, then we can form a topological module, denoted hˆB(M) over B . Applying
x x
this construction to B = z and M = Ω1(z ) we obtain a topological Lie-* algebroid G :=
g g crit
hˆzg(Ω1(z )) (see [CHA], Sect. 2.5.18 for details).
x g